On half-space problems for the weakly non-linear discrete Boltzmann
equation

Bernhoff, Niclas

doi:10.3934/krm.2010.3.195

Cited by 16 publications

(52 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…boundary layers [20,21,22,25]) it is preferable that the first component of the velocity is non-zero. However, the integer grid and the odd-integer grid are the same up to a shift and a scaling, and we could also use the integer grid as our basic universe.…”

Section: Examples Of Supernormal Dvmsmentioning

confidence: 99%

See 1 more Smart Citation

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Bernhoff¹

2016

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

Abstract. An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species as well as for binary mixtures. For binary mixtures also the concept of supernormal DVMs has been introduced by Bobylev and Vinerean. Supernormal DVMs are defined as normal DVMs such that both restrictions to the different species are normal as DVMs for single species.In this presentation we extend the concept of supernormal DVMs to the case of multicomponent mixtures and introduce it for polyatomic molecules. By polyatomic molecules we mean here that each molecule has one of a finite number of different internal energies, which can change, or not, during a collision. We will present some general algorithms for constructing such models, but also give some concrete examples of such constructions.The two different approaches above can be combined to obtain multicomponent mixtures with a finite number of different internal energies, and then be extended in a natural way to chemical reactions.The DVMs are constructed in such a way that we for the shock-wave problem obtain similar structures as for the classical discrete Boltzmann equation (DBE) for one species, and therefore will be able to apply previously obtained results for the DBE. In fact the DBE becomes a system of ordinary differential equations (dynamical system) and the shock profiles can be seen as heteroclinic orbits connecting two singular points (Maxwellians). The previous results for the DBE then give us the existence of shock profiles for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case.

show abstract

Section: Examples Of Supernormal Dvmsmentioning

confidence: 99%

“…The construction of the DVMs is such that also for half-space problems [17], as the Milne and Kramers problems [18], but also nonlinear ones [19], one obtain similar structures as for the classical discrete Boltzmann equation for one species [20,21,22,16].…”

Section: Introductionmentioning

confidence: 99%

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Bernhoff¹

2016

AIP Conference Proceedings

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, apart of this the general structure will be the same. We will below present some results for half-space problems that now can be extended to the case of multicomponent mixtures from the case of single species [5,6,9] (see also [7] for the case of binary mixtures).…”

Section: Boundary Layers For Mixturesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

“…We present the half-space problems and applicable existence results to our case, without any proofs, since they can be found elsewhere [5,6,9]. The results obtained in [6] can also be generalized by similar methods. To our knowledge no similar results exist in the continuous case for multicomponent mixtures, except for binary mixtures; for the linearized problem see [1], and for the nonlinear case, with equal masses, see [4].…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants

Bernhoff

Vinerean

2016

J Stat Phys

Self Cite

View full text Add to dashboard Cite

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

show abstract

Half-Space Problems for the Boltzmann Equation of Multicomponent Mixtures

Bernhoff

2022

Springer INdAM Series

View full text Add to dashboard Cite

On half-space problems for the weakly non-linear discrete Boltzmann equation

Cited by 16 publications

References 37 publications

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles

Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants

Half-Space Problems for the Boltzmann Equation of Multicomponent Mixtures

Contact Info

Product

Resources

About